Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
Подождите немного. Документ загружается.
9. (a) $5000 per page
(b) $1875 per page
(c) $625 per page
(d) $1750 per page
(e) No, no, no
11. 4 13. 7 15. .371
17. .417 19. 7 21. 2x 3 h
23. 3x
2
3xh h
2
25.
p(p
5
h)
27. (a) 49.2
(b) 48.1
(c) 48
(d) 48
29. Decreasing at
5
1
00
.
31. (a) They started at the same profit and ended at the same
profit.
(b) Dec 2008
33. (a) 10
(b) 15
(c) 5
35. (a) y 13.58x 184.95
(b) 13.58
(c) 12.8, 16.8. They are similar.
(d) 2010
Section 3.7, page 226
1. No 3. Yes 5. Yes
7. Yes 9. f
1
(x) x 11. f
1
(x)
x
5
4
13. f
1
(x)
3
x
2
5
15. f
1
(x)
x
2
4
7
17. f
1
(x)
1
x
19. f
1
(x)
1
2
x
x
21. f
1
(x)
3
x
5x
1
1
23. f (g(x)) x and g( f (x)) x
25. f (g(x)) x and g( f (x)) x
27. f (g(x)) x and g( f (x)) x
29. f ( f (x)) x
31. (a) 48
(b) 2
(c) 5
(d) 12
(e) Not enough information
(f) Not enough information
(g) Not enough information
986 ANSWERS
33.
35.
37.
39.
41.
x 0 43. x 0
45. (a) f
1
(x)
x
3
2
(b) f
1
(1)
3
1
,
f(
1
1)
1/5, therefore not equal
47. g(x)
x
m
b
49. (a) 1
(b) 1
(c) since the slopes are negative reciprocal of each other,
the lines are perpendicular to each other as well as sym-
metric with respect to the y x axis.
−1
1
2
−2
x
y
−2 −1
1 23
−10
−10
10
10
−10
0
10
10
x
y
51. f (g(x)) x and g( f (x)) x
53. True. Rotating the graph of an increasing function over
y x gives an increasing function.
Chapter 3 Review, page 229
1. (a) 3 (b) 1755 (c) 2 (d) 14
3.
5. Many possible answers, including these.
(a) If f (x) x 1, a 1, b 2, then f (a b)
f (1 2) f (3) 4, but f (a) f (b) f (1) f (2)
2 3 5, so the statement is false.
(b) If f (x) x 1, a 1, b 2, then f (ab) f (1
2)
f (2) 3, but f (a)
f (b) f (1)
f (2) 2
3 6, so
the statement is false.
7. r 4
9. t
2
t 2
11.
x
3
2
4
x 4
13. (a) f (t) 50 t
(b) g(t) 2500pt
(c) radius is 150 meters, area is 22,500p square meters
(d) after approximately 12.7 hours
15.
17.
(b)
19. No local maximum, local minimum at .5, increasing when
x .5, decreasing when x .5
21. Local maximum at 5.0704, local minimum at .263,
increasing when x 5.0704 and when x .263,
decreasing when 5.0704 x .263.
23.
−8
8
−66
−2
8642
6
4
2
8
x
y
ANSWERS 987
25. Here is one possibility:
27. Graph has symmetry with respect to x-axis, y-axis, and the
origin.
29. Even 31. Odd
33. Symmetric with respect to y-axis.
35.
37.
3 y 4
39. Many correct answers, including: any number from 2 to 3.5
or from 4.5 to 6
41. 1 43. 3 45. True
47. 4 49. x 3 51. x 3
53. (a) King Richard
(b) King Richard
(c) Fireball Bob
55. Shrink vertically by a factor of .25, then shift 2 units up.
57. Shift 7 units right, then stretch vertically by a factor of 3,
then reflect in the x-axis, and then shift 2 units up.
59. (e)
61. (a) 1/3
(b) (x 1)
x
2
5
(x 1)
(c)
(c
c
1
)
2
5
(c 0)
63.
−5
15
10
5
y
x
−5−10
10 15
4
x − 5
g(x) = 5 +
5
−2−3−4 −1
4321
3
2
1
−1
−2
−3
x
y
x 0124 tkb 11 b 6 2u
f(x) 753157 2t 7 2k 9 2b 5 2b 4u 5
x 4 3 2 101234
g(x) 143 11 3 2 4 3
h(x) g(g(x)) 3 3 4 314314
65. 82/27 67.
x
1
3
3
69. 1/4
71. (a)
x
2
1
1
(b)
x
1
2
1
73. x 0, x 1
75. (a) 1/3 (b) 5/8
77. 6 79. 3
81. 2x h
83. (a) $290 per ton
(b) $230 per ton
(c) $212 per ton
85. (a) approximately 45 to 50
(b) approximately 25 to 35
(c) approximately 30 to 44
87. (a) $5338; $9900; $6194 dollars per year
(b) $234,100
89. g(x) 5 (x 7)
2
x
2
14x 44; x 7
91.
93. The graph of f passes the horizontal line test and hence has
an inverse function. It is easy to verify either geometrically
[by reflecting the graph of f in the line y x] or alge-
braically [by calculating f( f(x))] that f is its own inverse
function.
95. There is no inverse function because the graph of f fails the
horizontal line test (use the viewing window with 10 x
20 and 200 y 100).
y
x
33
3
3
1
x
f(x) =
x
f
y
Inverse
988 ANSWERS
Chapter 3 Test, page 234
1. A(r) pr
2
2.
f(x) 3
1
3
f(x)
f(x)
3. (a) Domain: 12 x 12 or ⺢
(b) 2 y 1, 0 y 6
(c) 6
4. It is a function of x, because every input has exactly one
output. Answers invoking the vertical line test are also
acceptable.
y
2 1345
1
1
2
3
4
5
6
7
8
9
10
05234
x
y
2 1345
1
1
2
3
4
5
6
7
8
9
10
05234
x
y
2 1345
1
1
2
3
4
5
6
7
8
9
10
05234
x
5.
f(x h
h
) f(x)
2x
x
(x
2
x
h
)h
2h
x(x
2h
h)h
x(x
2
h)
6.
7.
(a) s(0) 25 If the child has read no books, the score will
be 25.
s
1
(35) 30 If the child wants a score of 35, s/he must
read 30 books.
(b) 80 books
8. Determine whether the given graph is symmetric with
respect to the y-axis, the x-axis, the origin, or none of
these:
(a) The origin
(b) The x axis
(c) None of these
9. Describe a sequence of transformations that will transform
the graph of the function f into the graph of the function g
f : Shift up by 2
g : Shift left by 5, and up by 9
10. The height in feet of a dropped ball after t seconds is given
by h(t) 16t
2
500.
(a) 97.6 feet/second
(b) 962 feet/second
(c) 96 feet/second
11. 0.63667 million shares/month
y
1.0 0.51.52.0
1.2
1.0
0.8
0.6
0.4
0.2
0.4
0.6
0.8
1.0
1.2
2.01.0 1.50.5
x
x(x
2
x
h)
2
x(
(
x
x
h
h
)
)
h
x
2
h
2
x
h
x
2
h
3
2
x
3
h
ANSWERS 989
12.
13.
f
°
g(x) f(g(x))
f(x
5
1)
5
(x
5
1
) 1
5
x
5
x
and
g
°
f(x) g( f(x))
g
5
x 1
5
x 1
5
1
x 1 1
x
Chapter 4
Section 4.1, page 247
1. Graph I 3. Graph K
5. Graph J 7. Graph F
9. Vertex: (5, 2). Since a 3, opens upward.
11. Vertex:
2
, p
. Since a 1, opens downward.
13. Vertex: (4, 3). Since a 2, opens upward.
15. Vertex:
3
2
,
4
5
. Since a 1, opens upward.
17. Vertex: (1, 3). Since a 4, opens downward.
19. Vertex: (0, 3). Since a 2, opens upward.
21. (a) 6x 3h 1
(b) Vertex:
1
6
,
1
1
2
(c) 3h
23. (a) 4x 2h 2
(b) Vertex:
1
2
,
1
2
(c) 2h
25. Rule: g(x) 2x
2
3. Vertex: (0, 3).
27. h(x) 2(x 5)
2
4
Vertex: (5, 4)
y
12
4
2
2
21
x
29. The rule is f(x) 3x
2
.
31. The rule is f(x) 2(x 3)
2
4.
33. f(x) (x 1)
2
4 35. f(x) x
2
6
37. b 0 39. b 4, c 8
41. x
11
2
1
, y
11
2
1
43. (a) $20,000
$100,000
(b) 200 units. Then
p(200) $110,000
(c) Approximately 365 units
45. The distance will be approximately 6,886 feet when the
height is 842 feet. The shell hits the water 13,987 feet, or
2.65 miles, away.
47. (a) B(30) 30 meters; B(100) 170 meters
(b) 50 kilometers per hour
49. 196 ft at 2.5 sec 51. 6.9 feet
53. b 15, h 15
55. When x 50 feet, then y 100 feet.
57. Maximum area is 704.2 square feet.
59. (a) y
11
1
52
x
2
(b) Thus x 34 inches
61. $4.00 63. $7.08
65. Minimum area: A
10,
p
000
3183.1
Section 4.2, page 257
1. This is a polynomial with leading coefficient 1, constant
term 1, and degree 3.
3. This is a polynomial with leading coefficient 1, constant
term 1, and degree 3.
5. This is a polynomial with leading coefficient 1, constant
term 3, and degree 2.
7. This is not a polynomial.
9. This is not a polynomial.
11. Quotient: 3x
3
3x
2
11x 17; remainder: 18
13. Quotient: x
2
2x 6; remainder: 7x 7
15. Quotient: x
2
2x 3; remainder: 0
17. Quotient: 5x
2
5x 5; remainder: 0
19. No 21. Yes
23. x {2, 5} 25. x {22
, 1}
27. x {3, 0} 29. The remainder is 2.
31. The remainder is 31.4375. 33. The remainder is 36.
35. The remainder is 183,424.
37. The remainder is 5,935,832p.
39. No 41. No
990 ANSWERS
43. Yes 45. Yes
47. (x 4)(2x 7)(3x 5)
49. (x 3)(x 3)(2x 1)
2
51. f(x) x
5
3x
4
5x
3
15x
2
4x 12
53. f(x) x
5
5x
4
5x
3
5x
2
6x
55. f(x) x
3
4x
2
25x 28
57. f(x) (x 1)(x 1)
59. f(x) (x 1)
2
(x 2)
2
(x p)
2
61. f(x) .25(x 8)(x 5)x
63. k 9 65. k 1
67. If x c were a factor, by the Factor Theorem, c would be a
root. Then c
4
c
2
1 0. This equation has discriminant
given by 1
2
4(1)(1) 3, hence it has no real roots.
Thus x c cannot be a factor for any real c.
69. (a) (x 2) is not a factor of x
3
2
3
.
(b) (c)
n
c
n
c
n
c
n
0.
71. k 5
73. 3, 2 45
, 2 45
Section 4.2.A, page 262
1. Quotient: 3x
3
2x
2
4x 1; remainder: 10
3. Quotient: 2x
3
x
2
3x 7; remainder: 29
5. Quotient: 5x
3
35x
2
242x 1690; remainder: 11836
7. Quotient: x
5
x
4
x
3
x
2
x 1; remainder: 0
9. Quotient: 3x
3
3
4
x
2
2
1
9
6
x
2
6
9
4
; remainder:
4
2
8
5
3
6
11. Quotient: x
2
2
x x 2
; remainder: 0
13. Quotient: x
2
x 30; remainder: 0
15.
x
1
2
(2x
4
6x
3
12x
2
10)
17. Quotient: x
2
4
2
3
0
x 4; remainder: 2.25
19. c 4
21. (a) x
4
x
3
2x
2
6x 21; remainder: 64
(b) When dividing by (x a) whenever a 3, the coeffi-
cients will be positive. When a 3, a 2 is positive
and a(a 2) is greater than 1.
(c) For a root a to be greater than three, the remainder when
dividing by (x a) would have to be zero. But in part b
we’ve shown it has to be a positive number.
Section 4.3, page 268
1. x 1, 1, 3 3. x 2, 1, 4
5. x
2
3
,
2
1
, 3 7. x 2, 0,
1
2
, 3
9. x 3, 2 11. x 1,
3
2
, 2
13. x 5, 2, 3 15. x 1
17. (x 1)(2x
2
3) 19. x
3
(x 4)(x
2
3)
21. (x 2)
3
(x
2
7)
23. Lower bound: 5; upper bound: 2
25. Lower bound: 1; no upper bound less than 3
27. Lower bound: 8; upper bound 4
29. x
2,
2
1
, 3
31. x
1
3
,
1
2
, 2
33. x
2,
37
2
5
,
(3
2
7 5)
35. x
1
2
, 3
, 2
, 3
, 2
37. x {1, 4, 3
, 3
}
39. x 50, x 2.24698, x 0.5549581,
x 0.80193774
41. (a) The only possible rational roots of x
2
2 are 1, 1,
2, 2. None of those numbers are roots of the polyno-
mial, so the square root of two is irrational.
(b) The only possible rational roots of x
2
3 are 1, 1,
3, 3. None of those numbers are roots of the polyno-
mial, so the square root of three is irrational.
(c) The only possible rational roots of x
2
4 are 1, 1,
2, 2, 4, 4. We try them all and find that 2 and 2
both are roots of the polynomial.
43. (a) 5.78 per 100,000 and 5.62 per 100,000
(b) Middle of 1997
(c) 1995
(d) 2004
(e) 2002–2004
45. The sides should be 2 inches.
47. (a) 6 degrees per day, 6.6435 degrees per day
(b) t 2.033 and t 10.7069 (days)
(c) t 5.0768 and t 9.6126
(d) t 4
49. (a) The carrying capacity is an upper bound on the popula-
tion of bunnies, while the threshold population is a
lower bound.
(b) The population is increasing
(c) It is decreasing
(d) It is decreasing
(e) kx(x T)(C x)
(f) x T, x C, x 0
Section 4.4, page 278
1. Yes 3. No 5. No
7. This could be the graph of a polynomial function of degree
at least 3 or 5.
9. Not the graph of a polynomial function.
11. This could be the graph of a polynomial function of degree
at least 5.
ANSWERS 991
13. In a very large window like 50 x 50, 100,000
y 1,000,000 the graph looks like the graph of y x
4
.
15. Roots 2, 1, 3; each has (odd) multiplicity 1.
17. Root 2 has multiplicity 1, root 1 has multiplicity 1, and
root 2 has (even) multiplicity 2 (or possibly higher).
19. (e) 21. (f) 23. (c)
25. Since this is a polynomial function of degree 3, the com-
plete graph must have another x-intercept.
27. Since this is a polynomial function of degree 4 with positive
leading coefficient, both of the ends should point up; here
they are pointing down.
29. 5 x 5, 50 y 30
31. 6 x 6, 60 y 320
33. 3 x 4, 35 y 20
35. 33 x 2, 50,000 y 260,000 then 2 x 3,
20 y 30
37. (a) The graph of a cubic polynomial can have no more than
two local extrema. If only one, the ends would go in the
same direction. Hence it can have two or none.
(b) When the end behavior and the number of extrema are
both accounted for, these four shapes are the only pos-
sible ones.
39. (a) The ends go in opposite directions, so the degree is odd.
(b) Since if x is large and positive, so is f (x), the leading co-
efficient is positive.
(c) Root 2 has multiplicity 1, root 0 has multiplicity
at least 2, root 4 has multiplicity 1, root 6 has multi-
plicity 1.
(d) Adding the multiplicities from c, the polynomial must
have degree at least 5.
41.
43.
x intercepts: 1, 1 3
;
local max: (2, 2);
local min: (0, 2)
1122345
20
40
40
20
x
3
y
x
y
45. x intercepts: 2, 3;
local min:
1
4
1
,
2
2
5
7
6
47. x intercepts:
1
2
5
, 0; local max: (1.30, 1.58)
(.345, .227); local min: (.345, .227)(1.30, 1.58)
49. (a)
(b) 4950, 3750. The rate of increase is decreasing.
(c) After this point, you get less and less benefits for a unit
of expenditure.
51. 6 additional trees per acre
53. r 3.046, h 12.18
10
0
160000
120000
80000
40000
x
20 30 40
y
11
5
10
10
5
x
2
y
1
0
2
0.5
1
x
3
1.5
2
y
992 ANSWERS
55. It is a good approximation for 3 x 3.
57. It is a good approximation for 3 x 2.25.
59. (a)
If the graph had the horizontal portion appearing in the
window, then the equation g(x) 4 would have infinitely
many solutions, which is impossible, since a polynomial
equation of degree 3 can have at most three solutions.
(b) 1 x 3 and 3.99 y 4.01
(c) If the graph of a polynomial of degree n had a horizon-
tal portion appearing in the window, it would be a por-
tion also of the line y k. Then the equation f(x) k
would have infinitely many solutions, which is impossi-
ble, since a polynomial equation of degree n can have at
most n solutions.
6
3
3
0
2
200
x
46 1354623 51
100
300
400
y
2
4
2
x
46642
4
2
y
61. (a)
(b)
(c)
(d) Different windows would be needed for different por-
tions of the graph. 20 x 3 and
5 10
6
y 10
6
; then 3 x 2 and
5000 y 60,000; then 1 x 5 and
5000 y 5000; finally 5 x 11 and
100,000 y 100,000.
Section 4.4.A, page 286
1. A cubic model seems best.
3. A quadratic model seems best.
5. (a) f(x) 1.239x
3
26.221x
2
19.339x 4944.881
(b) 4055.7, 3633.7
(c) Very accurate
(d) The model predicts a large surge in crime after 2005.
The data do not support this conclusion. So the model
should not be used to predict data after 2005.
7. (a)
−10
10
−19 11
−10
10
−10 10
20
x
18 2 10
y
20
ANSWERS 993
(b)
(c) Evaluating the function in (a) gives:
noon (x 12) 80.4°, 9 A.M.(x 9) 69.1°, 2 P.M.
(x 14) 82.8°.
9. (a)
(b) cubic model f(x) 14.847x
3
321.847x
2
1444.238x 42266.864
(c) $8,181
(d) The model predicts that the dropoff in median income
that began in 2002 will continue, and get worse and
worse. The reasonability of the model depends on your
opinion of this prediction.
11. (a)
(b) f(x) .340x
2
4.410x 22.092,
f(x) .0156x
3
.0115x
2
2.55x 24.39
f(x) .00746x
4
.224x
3
1.896x
2
3.684x 29.531
(c) The quartic model fits the data best, but given the down-
ward trend in smoking, the cubic model is probably best
for the future.
x
1
36
32
28
24
3579
11 13
y
x
20
40000
46000
45000
44000
42000
41000
43000
46810
12 14
y
13. (a) Cubic and quartic models are shown and graphed:
(b) Both models seem to rise disturbingly steeply and only
a pessimist would expect them to be accurate very far
into the future.
Section 4.5, page 300
1.
∞,
4
3
<
4
3
, ∞
i.e. x Å
4
3
3. All real numbers
5. (∞, 3) < (3, 1) < (1, 3) < (3, ∞)
7. f(x)
x
2
2
x
2
9
9. f(x)
x
4
x
1
4
0
x
2
3
6
9
11. F 13. A
15. Vertical asymptotes at x 0 and x 1
46 2
2
2 4 6
2
x
4
6
y
x
4
36
(x 3)(x 1)(x 1)(x 3)
4 2264108
2
4
4
2
x
108 6
y
994 ANSWERS
17. Vertical asymptotes at x
7
2
41
19. Horizontal asymptote: y 0. All viewing windows will
work.
21. Horizontal asymptote: none
23. Horizontal asymptote: y
2
3
. 25 x 25, 5 y 5
25. Vertical asymptote: x 2
Horizontal asymptote: y 0
y-intercept:
1
2
27. Vertical asymptote: x 1
Horizontal asymptote: y 2
y-intercept: 0
29. Vertical asymptotes: x 2, x 3
Horizontal asymptote: y 0
Hole:
0,
1
6
y-intercept: none
112345
2
4
4
2
x
6
8
y
1122345
2
4
4
2
x
345
6
6
y
1122345
2
4
4
2
x
345
y
31. Horizontal asymptote: y 0
y-intercept: 2
33. Vertical asymptotes: x 5, x 1
Horizontal asymptotes: y 0
y-intercept:
1
5
35. Vertical asymptotes: x 2, x 1
Horizontal asymptotes: y 2
Holes:
2,
9
4
y-intercept:
1
2
37. Three windows are needed: 5 x 4.4 and 8 y
4; then 2 x 2 and .5 y 5; finally, 15
x 3 and .07 y .02
39. 9.4 x 9.4 and 4 y 4
41. 5 x 15, 2 y 2
43.
x(x
1
h)
45.
(x 2)(x
3
h 2)
47. 3
x
2
2
(x
x
h
h
)
2
49. (a) 1/4 (b) 1/9 (c) They are identical.
51. (a) g(x)
x
x
1
2
(b) g(x)
x
x
1
2
2 4 2 6
2
4
6
4
2
x
46
y
11223
2
4
6
4
6
2
x
345678
y
1122346
1
x
346 55
2
y
ANSWERS 995
(c)
53. 8.4343 in. by 8.4343 in. by 14.057 in.
55. (a) 10
(b) 17
(c) Number of figures
2
20
.2
5
5
t
t
(d)
(e) He will be able to buy 18 in the year 2012. He will never
be able to buy 21.
57. (a) x
40
x
0
(b) 30 10 5
x 30 105
or (7.6 x 52.4)
(c) 20, 20 10
59. (a) h
1
h 2
(b) h
1
150/pr
2
2
(c) p(r 1)
2
(150/pr
2
2)
(d) Otherwise (r 1) would be negative.
(c) r 2.879, h 5.759
61. (a) 9.801 m/s
2
(b)
(c) No
0
10
0 9,000,000
x
0
20
15
10
5
0 102030
y
6
4.7
4.7
6